L² operator

The L² operator stands for the squared magnitude of angular momentum — the total amount of rotation, regardless of direction. It is assembled from the three component operators as L² = Lx² + Ly² + Lz², and unlike the individual components it does commute with all of them. That makes L² special: you can know the total amount of angular momentum at the same time as one of its components, even though the three components themselves cannot all be known together.

When L² acts on an angular-momentum state, it returns the value ℓ(ℓ+1)·ℏ². The whole number ℓ is the azimuthal quantum number, so measuring L² is effectively reading off ℓ. The peculiar form ℓ(ℓ+1) rather than a plain ℓ² is not a typo; it emerges directly from the algebra of the operators and reflects the fact that the perpendicular components, though uncertain, still contribute to the total.

Because it commutes with the energy operator in any spherically symmetric problem, L² is a conserved quantity there: the total angular momentum of an electron orbiting a nucleus does not change over time. This is why ℓ is a good, stable label for atomic states, sitting alongside the principal quantum number n in describing every orbital.